| L(s) = 1 | + 1.59·2-s + 0.551·4-s − 2.02·5-s − 1.90·7-s − 2.31·8-s − 3.23·10-s + 6.42·11-s + 0.315·13-s − 3.04·14-s − 4.79·16-s + 0.873·17-s − 4.25·19-s − 1.11·20-s + 10.2·22-s + 23-s − 0.908·25-s + 0.504·26-s − 1.05·28-s − 29-s + 2.71·31-s − 3.04·32-s + 1.39·34-s + 3.86·35-s − 8.06·37-s − 6.79·38-s + 4.67·40-s + 2.41·41-s + ⋯ |
| L(s) = 1 | + 1.12·2-s + 0.275·4-s − 0.904·5-s − 0.721·7-s − 0.817·8-s − 1.02·10-s + 1.93·11-s + 0.0875·13-s − 0.815·14-s − 1.19·16-s + 0.211·17-s − 0.975·19-s − 0.249·20-s + 2.18·22-s + 0.208·23-s − 0.181·25-s + 0.0988·26-s − 0.199·28-s − 0.185·29-s + 0.488·31-s − 0.537·32-s + 0.239·34-s + 0.652·35-s − 1.32·37-s − 1.10·38-s + 0.739·40-s + 0.377·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.214954602\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.214954602\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 1.59T + 2T^{2} \) |
| 5 | \( 1 + 2.02T + 5T^{2} \) |
| 7 | \( 1 + 1.90T + 7T^{2} \) |
| 11 | \( 1 - 6.42T + 11T^{2} \) |
| 13 | \( 1 - 0.315T + 13T^{2} \) |
| 17 | \( 1 - 0.873T + 17T^{2} \) |
| 19 | \( 1 + 4.25T + 19T^{2} \) |
| 31 | \( 1 - 2.71T + 31T^{2} \) |
| 37 | \( 1 + 8.06T + 37T^{2} \) |
| 41 | \( 1 - 2.41T + 41T^{2} \) |
| 43 | \( 1 + 1.70T + 43T^{2} \) |
| 47 | \( 1 - 6.56T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 - 9.53T + 67T^{2} \) |
| 71 | \( 1 - 11.9T + 71T^{2} \) |
| 73 | \( 1 + 1.07T + 73T^{2} \) |
| 79 | \( 1 - 7.64T + 79T^{2} \) |
| 83 | \( 1 - 1.98T + 83T^{2} \) |
| 89 | \( 1 + 3.76T + 89T^{2} \) |
| 97 | \( 1 - 9.41T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.139886723301909589762549030757, −6.95252905961396854579639088129, −6.62692343620746317597658995875, −5.96815151019489112475465493618, −5.06136968607780534310778230157, −4.19734851028471397125785535033, −3.77346890911379411684336289114, −3.30400064093727752826380487557, −2.08760274225940681182729160530, −0.64855764897323813365083991831,
0.64855764897323813365083991831, 2.08760274225940681182729160530, 3.30400064093727752826380487557, 3.77346890911379411684336289114, 4.19734851028471397125785535033, 5.06136968607780534310778230157, 5.96815151019489112475465493618, 6.62692343620746317597658995875, 6.95252905961396854579639088129, 8.139886723301909589762549030757
